The state $$|\Psi \rangle = |0\rangle + \sum_j \int d\omega f_j(\omega)\hat{a}^\dagger_j (\omega) |0\rangle $$ is coming from a far field and incident on a double slit setup. Here j is the index of j-th atom of the detector and $\omega$ is the index of the $\omega$-th mode of the field.

On the other side there is a detector. The setup looks as follows (Prof. Shih's book).

So far, I have calculated the following. The field at the detector is described as follows.